8.2 Further developments8 Conclusion8 Conclusion

8.1 Evaluation of the methods

An assessment of the quality of the numerical methods should consider, at least, the following aspects: (i) accuracy and robustness in describing high Lorentz factor flows with strong shocks; (ii) effort required to extend to multi dimensions; (iii) effort required to extend to RMHD and GRHD. In Table  10 we have summarized these aspects of numerical methods for SRHD.
Method Ultra-relativistic regime Handling of discontinuities Popup Footnote Extension to several spatial dimensions Popup Footnote Extension to
GRHD RMHD
AV-mono tex2html_wrap_inline6539 O, SE tex2html_wrap_inline6541 tex2html_wrap_inline6541 tex2html_wrap_inline6541
cAV-implicit tex2html_wrap_inline6541 tex2html_wrap_inline6541 tex2html_wrap_inline6539 tex2html_wrap_inline6539 tex2html_wrap_inline6539
HRSC Popup Footnote tex2html_wrap_inline6541 tex2html_wrap_inline6541 tex2html_wrap_inline6541 Popup Footnote tex2html_wrap_inline6541 Popup Footnote tex2html_wrap_inline6539 Popup Footnote
rGlimm tex2html_wrap_inline6541 tex2html_wrap_inline6541 tex2html_wrap_inline6539 tex2html_wrap_inline6539 tex2html_wrap_inline6539
sTVD tex2html_wrap_inline6541 Popup Footnote D tex2html_wrap_inline6541 tex2html_wrap_inline6541 tex2html_wrap_inline6541
van Putten tex2html_wrap_inline6541 Popup Footnote D tex2html_wrap_inline6541 tex2html_wrap_inline6539 tex2html_wrap_inline6541
FCT tex2html_wrap_inline6541 O tex2html_wrap_inline6541 tex2html_wrap_inline6539 tex2html_wrap_inline6539
SPH tex2html_wrap_inline6541 D, O tex2html_wrap_inline6541 tex2html_wrap_inline6541 Popup Footnote tex2html_wrap_inline6539 Popup Footnote
  
Table 10: Evaluation of numerical methods for SRHD. Methods have been categorized for clarity.

Since their introduction in numerical RHD at the beginning of nineties, HRSC methods have demonstrated their ability to describe accurately (stable and without excessive smearing) relativistic flows of arbitrarily large Lorentz factors and strong discontinuities, reaching the same quality as in classical hydrodynamics. In addition (as it is the case for classical flows, too), HRSC methods show the best performance compared to any other method (e.g., artificial viscosity, FCT or SPH).

Despite of the latter fact, a lot of effort has been put into improving these non-HRSC methods. Using a consistent formulation of artificial viscosity has significantly enhanced the capability of finite difference schemes [131Jump To The Next Citation Point In The Article] as well as of relativistic SPH [164Jump To The Next Citation Point In The Article] to handle strong shocks without spurious post-shock oscillations. However, this comes at the price of a large numerical dissipation at shocks. Concerning relativistic SPH, recent investigations using a conservative formulation of the hydrodynamic equations [30Jump To The Next Citation Point In The Article, 164] have reached an unprecedented accuracy with respect to previous simulations, although some issues still remain. Besides the strong smearing of shocks, the description of contact discontinuities and of thin structures moving at ultra-relativistic speeds needs to be improved (see Section  6.2).

Concerning FCT techniques, those codes based on a conservative formulation of the RHD equations have been able to handle relativistic flows with discontinuities at all flow speeds, although the quality of the results is below that of HRSC methods in all cases [161Jump To The Next Citation Point In The Article].

The extension to multi-dimensions is simple for most relativistic codes. Finite difference techniques are easily extended using directional splitting. Note, however, that HRSC methods based on exact solutions of the Riemann problem [109Jump To The Next Citation Point In The Article, 187Jump To The Next Citation Point In The Article] first require the development of a multidimensional version of the relativistic Riemann solver. The adapting-grid, artificial viscosity, implicit code of Norman & Winkler [131] and the relativistic Glimm method of Wen et al. [187] are restricted to one dimensional flows. Note that Glimm's method produces the best results in all the tests analyzed in Section  6 .

The symmetric TVD scheme proposed by Davis [38Jump To The Next Citation Point In The Article] and extended to GRMHD (see below) by Koide et al. [82Jump To The Next Citation Point In The Article] combines several characteristics making it very attractive. It is written in conservation form and is TVD, i.e., it is converging to the physical solution. In addition, it is independent of spectral decompositions, which allows for a simple extension to RMHD. Quite similar statements can be made about the approach proposed by van Putten [181Jump To The Next Citation Point In The Article]. In contrast to FCT schemes (which are also easily extended to general systems of equations), both Koide et al.'s and van Putten's methods are very stable when simulating mildly relativistic flows (maximum Lorentz factors tex2html_wrap_inline6611) with discontinuities. Their only drawback is an excessive smearing of the latter. A comparison of Davis' method with Riemann solver based methods would be desirable.



8.2 Further developments8 Conclusion8 Conclusion

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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